Using simulation based on rnorm, study the performance of unbiased estimator of standard deviation (using n-1 in denominator) and two nearest alternatives that use n or n-2 in the denominator instead. Go over samples ranging in size from 2 to 20; for each sample size n draw multiple (say, 1000) samples of that size from normal distribution with zero mean and standard deviation of one; for each sample calculate

(a) standard deviation within each such sample;

(b) biased sd alternative that uses n instead of n-1;

c) sd alternative that uses n-2 (write functions if needed to calculate what you need); at this point, for each sample you have three estimators calculated: sd, sd.n, sd.n.1. Remember that they are functions of the (random) sample they are calculated for, and hence they are random variables themselves. We want to know how well each of them approximates the true underlying standard deviation (which we happen to know, because we simulate by drawing from a known, defined distribution). In order to accomplish this, calculate the mean and spread of each of these three standard deviation estimates across all samples. Plot the calculated means and spreads of all three estimators of the standard deviation as functions of sample size n, observe how they approach standard deviation of the underlying distribution (we know it is 1 since this is what we used to generate the data!), describe your observations in plain English. (Hints: R function points may prove useful: remember also that you can use plot with parameter type=n to create blank plot of the size required for the data to fit and then add data series one at a time using points).